Fall Sem. 2023-24
Programme Name & Branch: B. Tech.
Course Name & Code: Complex Variables and Linear
Algebra (BMAT201L)
Practice all the questions
 xy 2 ( x  iy )
, if z  0

1. Show that the function f ( z )   x 2  y 4
0,
if z  0

is not analytic at origin although C-R equations are satisfied at the origin.
Hint: Verify along the path x  my 2 .
2. Determine whether the function f ( z )  z z is analytic at z  0.
3. If w  f ( z )  x 2  ay 2  2 xy  i (bx 2  y 2  2 xy ) is analytic, then find the values of a and b.
Answer: a  1, b  1.
4. Find
the
analytic
function
f ( z )  u  iv
in
terms
z of whose
real
part
is
u  e2 x ( x cos 2 y  y sin 2 y) an analytic function of z and hence find v.
2z
Answer: f ( z )  z e  c and v  e2 x ( x sin 2 y  y cos 2 y )
5. If w  f ( z )    i represents the complex potential function for an electric field and
 ( x, y )  x 2  y 2 
x
, then find  ( x, y).
x  y2
2
Answer:  ( x, y )  2 xy 
y
k
x  y2
2